with Exact Dependence on the c-Quark Mass Abd

121 Pages·2015·1.61 MB·English

Strong Interaction Corrections to the Weak
Radiative B-Meson Decay at Order O(α2) with
s
Exact Dependence on the c-Quark Mass
Abdur Rehman
PhD dissertation under the supervision of
prof. dr hab. Miko(cid:32)laj Misiak
at the Institute of Theoretical Physics, Faculty of Physics, University of Warsaw.
Warsaw, July 2015

Abstract
¯
The process B → X γ is known to provide important constraints on extensions of
s
the Standard Model (SM). The present SM prediction for its CP- and isospin-averaged
branching ratio reads B(B¯ → X γ)SM = (3.36±0.23)·10−4. It agrees very well with the
s
current experimental average B(B¯ → X γ)exp = (3.43±0.22)·10−4. The experimental
s
accuracy is expected to improve in a significant manner after the Belle-II experiment
begins collecting data within the next few years. Consequently, theoretical calculations
must also be upgraded to match the experimental precision.
A considerable contribution to the current theoretical uncertainty originates from
the fact that some of the Next-to-Next-to-Leading-Order strong interaction corrections
(called K(2) and K(2)) have not yet been calculated for an arbitrary value of the charm
17 27
and bottom quark mass ratio m /m . Instead, known results for these corrections at
c b
m = 0 and for m (cid:29) m /2 serve as a basis for an interpolation in m , which introduces
c c b c
around ±3% uncertainty into B(B¯ → X γ)SM.
s
Inordertoremovethisuncertainty,determiningtheexactdependenceofK(2) andK(2)
17 27
on the c-quark mass is necessary. In the language of Feynman diagrams with unitarity
cuts, four-loop diagrams with two mass scales (m and m ) need to be evaluated. The
c b
necessary ultraviolet counterterms involve three-loop two-mass-scale diagrams that must
be calculated up to O(ε) in the dimensional regularization parameter ε.
In the present thesis, we evaluate [1] the exact dependence on the c-quark mass of
all the necessary ultraviolet-counterterm diagrams that contribute to the yet-unknown
parts of K(2) and K(2). These corrections originate from interferences of four-quark and
17 27
photonic dipole operators. They are currently responsible for the main uncertainty in the
perturbative contribution to B(B¯ → X γ)SM.
s
Apart from the calculation for arbitrary m , we also evaluate many of the necessary
c
counterterm contributions at m = 0, and present them to all orders in ε wherever
c
possible. Our results have contributed to the evaluation of the m = 0 boundary for the
c
interpolation, and thus to the recently published updated phenomenological analysis of
B(B¯ → X γ)SM [2].
s
The thesis contains many technical details that have not been presented elsewhere,
namely explicit expressions for all the relevant quantities in terms of the master integrals,
as well as results for these integrals obtained using several different methods, involving
Mellin-Barnes techniques and differential equations.
3

Contents
1 Introduction 6
¯
2 Inclusive B → X γ in the Standard Model 14
s
2.1 Theoretical framework for radiative B-decays . . . . . . . . . . . . . . . 14
2.1.1 The effective Lagrangian and choice of the operators basis . . . . 15
2.1.2 Determination and renormalization of the Wilson coefficients . . . 19
2.1.3 Matrix elements and the role of m . . . . . . . . . . . . . . . . . 27
c
2.2 Renormalization of the matrix elements and the NNLO counterterms for
arbitrary m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
c
2.3 The CP- and isospin-averaged branching ratio . . . . . . . . . . . . . . . 37
¯
2.3.1 Theoretical uncertainties in the SM prediction for B(B → X γ) . 42
s
3 A description of the calculational methods 45
3.1 Feynman integrals and methods of their evaluation . . . . . . . . . . . . 46
3.2 Integration by parts, reverse unitarity and reduction to master integrals . 48
3.2.1 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.2 Reverse unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.3 Reduction to master integrals . . . . . . . . . . . . . . . . . . . . 51
3.3 Evaluation of the master integrals . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 The Feynman and Schwinger parameterizations . . . . . . . . . . 56
3.3.2 The Mellin-Barnes method . . . . . . . . . . . . . . . . . . . . . . 60
3.3.3 Differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Results for the NNLO QCD counterterm contributions 78
4.1 Final results for an arbitrary charm quark mass . . . . . . . . . . . . . . 79
4.2 Results for m = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
c
4.3 Plots and their interpretation . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Exact coefficients at the master integrals . . . . . . . . . . . . . . . . . . 91
4.4.1 Coefficients for an arbitrary charm quark mass . . . . . . . . . . . 91
4.4.2 Coefficients for the case of a vanishing charm quark mass . . . . . 93
5 Outlook: bare NNLO QCD contributions for arbitrary m 96
c
6 Conclusions 99
5
Chapter 1
Introduction
This thesis is based on the Standard Model (SM) [3–18] of elementary particle physics
which describes our current knowledge of the three fundamental non-gravitational forces:
strong, electromagnetic and weak. It is remarkable that all these three interactions are
based on a common principle of gauge invariance, and successfully described by a local
relativistic and renormalizable quantum field theory.1 The mechanism of Electroweak
Symmetry Breaking (EWSB) occurring at the scale v ∼ 248 GeV is a main component
of the SM. The scale v corresponds to the vacuum expectation value of the Higgs doublet
that gives masses to the weak bosons W± and Z0, leaving the spinless boson h0 as the
physical degree of freedom. In July 2012, the SM emerged as a complete theory in
the sense that its last missing particle h0 was experimentally found with mass around
126 GeV [20,21].
The SM is an impressive theoretical achievement and one of the best successfully
tested theories of contemporary physics. It has been investigated to a great precision at
dedicated particle accelerator facilities [22]. In particular, the gauge sector of the SM has
been extensively studied at the Large Electron-Positron (LEP) collider [23] at the Eu-
ropean Organization for Nuclear Research (CERN), at the Stanford Linear Accelerator
Center (SLAC), as well as at the Tevatron accelerator at the Fermi National Accelerator
Laboratory (FNAL). These experiments have tested many SM observables reaching the
accuracy of below per-mille level, becoming sensitive to loop quantum corrections of elec-
troweak origin. Also, loop corrections due to Quantum Chromodynamics (QCD) played
an important role in comparing theory predictions to the experimental results. Some of
the noteworthy examples of relevant higher-order electroweak corrections occur in the
case of the ρ-parameter [24–28], the muon decay and the Fermi constant G [29,30], the
F
Weinberg mixing angle extracted from leptonic observables [31], as well as anomalous
magnetic moments of both the electron and the muon. In the anomalous magnetic mo-
ment cases, accuracies reaching one part in 109 [32] and one part in 106 [22] have been
reached for the electron and muon, respectively. With such an accuracy, the anomalous
magnetic moments (despite being leptonic quantities) become sensitive to strong inter-
action effects, which in the muonic case are the main limitation for further improving
the accuracy on the theory side. Other examples of observables allowing for very precise
comparisons between the SM predictions and data are the bound state spectra of positro-
nium [33] and muonium. At present, no clear contradiction between the SM predictions
1 See Ref. [19] for a sample list of textbooks on the SM and field theory.
6
and experimental data is observed, after supplementing the SM with neutrino masses
stemming from dimension-five operators, as well as an extra weakly-interacting particle
to describe Dark Matter (DM) in the Universe. There are a few measurements in which
tensions with the SM predictions occur, but they either seem to be debatable or at least
statistically allowed given the large number of observables considered.
Despite its success, one should keep in mind that the SM is likely only an effective
theory that describes Nature at low energies, at or below the EWSB scale. At higher
energies, beyond-SM degrees of freedom may become dynamical. Their possible existence
could help us in better understanding the EWSB mechanism, neutrino masses and mix-
ings, or the observed asymmetry between baryons and anti-baryons in the Universe. In
general, a New Physics (NP) theory at the TeV scale or above is expected to satisfy the
following requirements: (i) its gauge group should contain the SU(3) ×SU(2) ×U(1)
C L Y
of the SM, (ii) it should incorporate all the SM degrees of freedom either as fundamental
or composite fields, and (iii) it should reduce to the SM in the low-energy limit. A search
for Beyond-SM (BSM) theories is actively being carried out in two complementary ways,
namely via direct production searches (high energy frontier) and via indirect searches
(high intensity frontier). Both approaches require sincere theoretical predictions with
quantifiable error estimates. In both of them, an important issue is to keep quantum
effects under control, particularly the QCD ones. The main purpose of the Large Hadron
Collider (LHC) are the direct searches at the TeV scale and beyond. So far, no direct
evidence of NP has been found. At the same time, low energy measurements are becom-
ing progressively accurate, increasing their potential of indirect searches. This requires
higher precision on the theoretical side. In many cases, higher-order perturbative quan-
tum corrections need to be calculated. It is particularly relevant for processes where
virtual exotic particles might contribute to loop amplitudes. A well-known class of such
processes are Flavor Changing Neutral Current (FCNC) decays which arise only at the
one-loop level in the SM. An important difficulty in their case is that they involve quarks,
so a good control over QCD effects is required to reach a percent-level accuracy.
The flavor structure of the SM is dictated by the Higgs-quark-antiquark Yukawa in-
teractions which generate the quark masses when the Higgs field acquires its vacuum
expectation value. The Yukawa coupling matrices contain a sizeable number of indepen-
dent parameters. In the quark sector, these are the 6 physical masses of quarks and four
parameters (3 angles and one phase) of the Cabibbo-Kobayashi-Maskawa (CKM) ma-
trix [34,35]. The CKM matrix describes the quark mass eigenstate mixing under weak
interactions. NumericalvaluesoftheseparametersarenotpredictedbytheSMbutrather
have to be extracted from measurements before making any theoretical prediction.2
In the present thesis, we will focus on the quark flavor sector of the SM in the context
of a subclass of B-meson decays, specifically the weak radiative B-meson decay. For
definiteness, let us consider the mesons being bound states of the b quark and one of the
light antiquarks (u¯ or d¯). The bu¯ bound state is called B−, and the bd¯bound state is
called B¯0. Both of them will commonly be denoted by B¯. We shall consider their weak
radiative decays into a photon and any hadronic final state that contains no charmed
(C (cid:54)= 0) particles, and has nonvanishing strangeness (S (cid:54)= 0), i.e. contains an unbalanced
s quark. We shall sum over all the possible final states satisfying the above requirements,
2 Some standard reviews of heavy flavor physics can be found in Refs. [36,37].
7
which means considering an inclusive process. The branching ratio of this process is
(cid:0)¯ (cid:1)
denoted by B B → X γ . Its evaluation and/or measurement involves taking an average
s
over B¯ = B− and B¯ = B¯0 (see Sec. 2.3 for more details).
¯
Since the early 1990’s, the decay B → X γ has been one of the most frequently con-
s
sidered processes in flavor physics. It is well known as an invaluable and-well established
means to constrain parameter spaces of BSM models. Being generated by the quark-level
b → sγ FCNC transition in the SM, it receives dominant contributions from loop dia-
grams involving the W boson and up-type quarks. Sample Leading Order (LO) diagrams
¯
for B → X γ in the SM, multi-Higgs doublet models and the Minimal Supersymmetric
s
Standard Model (MSSM) are shown in Fig. 1.1. One can see that the SM contribution
¯
Figure 1.1: Sample LO diagrams for B → X γ in the SM (a), multi-Higgs doublet model
s
(b) and MSSM (c), respectively.
is of the same perturbative order as the possible BSM ones. Comparable contributions
in the multi-Higgs doublet models can arise from loops with charged scalars as shown in
Fig. 1.1(b). In the supersymmetric theories, chargino-squark loops shown in Fig. 1.1(c)
often become important, even in scenarios with minimal flavor violation and moderate
tanβ. In the SM, the considered decay receives an additional chirality suppression by
a factor of m /m . Such a chirality suppression may be off-set in certain NP models
b W
like the MSSM with large tanβ or left-right models. Therefore, constraints from b → sγ
¯
on such models are particularly severe [38,39]. However, the power of B → X γ for
s
indirectly testing NP models depends on the accuracy of its measurements and precision
of theoretical predictions.
¯
AnotherapplicationofB → X γ thathasfrequentlybeendiscussedintheliteratureis
s
constraining the Heavy Quark Effective Theory (HQET) [40] parameters that matter for
extraction of the CKM elements |V | and |V | from the semileptonic B-meson decays.
cb ub
However, this application is now mostly of historical importance because of growing
accuracyinthedeterminationoftheseparametersfromthesemileptonicdecaysalone[41],
¯
and to intrinsic uncertainties generated by the charm-quark loop contributions to B →
X γ.
s
As far as the measurements are concerned, the first observation of an exclusive
hadronic process that is generated by b → sγ, namely B → K∗γ, was performed by
the CLEO collaboration in 1993 [42]. Both at CLEO and at the so-called B-factories
that started their operation in the late 1990’s (Belle and Babar), electron-positron col-
lisions at the center-of-mass energy overlapping with the Υ(4S) resonance were used to
produce the B mesons. At Belle, a significant fraction of time was also spent at the
Υ(5S) resonance, where B production in addition becomes kinematically allowed.
s
¯
The current measurements of the CP- and isospin-averaged branching ratio of B →
X γ performedbyCLEO[43],Belle[44,45]andBabar[46]-[50]contributetothefollowing
s
8
(a) (b)
¯
Figure 1.2: The measured photon energy spectra in B → X γ as measured by the (a)
s
Babar [46] and (b) Belle [44] experiments. The peaks are centered around E (cid:39) m /2 (cid:39)
γ b
2.35GeV which corresponds to the photon energy in the two-body partonic decay b → sγ
with an approximately massless s quark.
world average [51]
B(cid:0)B¯ → X γ(cid:1)exp = (3.43±0.21±0.07)·10−4, (1.1)
s (E0=1.6)
wherethelasterror(±0.07)originatesfromthephotonspectrummodeling, whilethefirst
one (±0.21) contains the remaining systematic errors together with the statistical one.
This average, performed by the Heavy Flavor Averaging Group (HFAG), corresponds to
including only photons whose energies E are larger than E in the B-meson rest frame,
γ 0
and E is set to 1.6GeV. The averaging involves an extrapolation3 from measurements
0
performed at E ∈ [1.7,2.0]GeV. A combination of the experimental results and their
0
extrapolation to E = 1.6GeV are performed in the same step, to minimize model depen-
0
dence. The experimental cuts at E ∈ [1.7,2.0]GeV are necessary due to rapidly growing
0
backgrounds at lower energies. On the other hand, theoretical predictions become less
precise with growing E (see below). Thus, an intermediate value of E must be chosen
0 0
for comparing theory with experiment. A conventional choice of E = 1.6GeV was pro-
0
posed in Ref. [52], and it is being followed since then. The raw photon energy spectra
in the inclusive measurements are shown in Fig. 1.2. One can see that the uncertainties
grow for smaller photon energies in both plots. It is due to subtraction of a larger and
more uncertain background. The background originates from the so-called continuum
processes (i.e. e+e− collisions that produce no B mesons), as well as, e.g., radiatively
decaying π0 and η particles produced in purely hadronic B-meson decays.
It is worth to emphasize that at the LHCb (a hadronic collider) at CERN, only
measurementsofexclusiveb → sγ decaymodesarefeasible. TheadvantageofB-factories
is that radiative B decays can be studied both inclusively and exclusively. An important
(cid:0)¯ (cid:1)
improvement in the accuracy of the inclusive B B → X γ measurement is expected at
s
Belle II [53] which is being constructed at present, and scheduled to begin collecting data
in October 2017. It aims at collecting 50 times more e+e− → Υ(4S) → BB¯ events than
3 Further comments on this extrapolation can be found in Sec. 2.1.3.
9
(cid:0)¯ (cid:1)
Figure 1.3: A summary of the B B → X γ measurements, and their comparison to the
s
current SM prediction [2].
the previous B-factories taken together. Much larger statistics will allow to efficiently
use the so-called hadronic tagging which practically removes the continuum background,
and thus leads to a reduction of systematic errors, too.
On the theoretical side, the flavor changing weak interactions are described by the
well-established Effective Field Theory (EFT) framework. Exploiting the Heavy Quark
Expansion (HQE) within this framework, one shows [54–61] that the inclusive decay rate
Γ(cid:0)B¯ → X γ(cid:1) is well approximated by the partonic decay rate4
s
Γ(cid:0)B¯ → X γ(cid:1) = Γ(cid:0)b → Xpartonicγ(cid:1)+δΓ . (1.2)
s s nonp
Here, Xpartonic stands for s,sg,sgg,sqq¯,... being partonic states with q = u,d,s only.
s
(cid:0)¯ (cid:1)
The detailed analysis of Ref. [61] results in an estimate of δΓ /Γ B → X γ at the
nonp s
level of 0.05 or below, which is often expressed as the statement that “nonperturbative
corrections” do not exceed 5% of the decay rate. The qualitative reason why the consid-
ered inclusive decay of the B meson can be very well described perturbatively is the fact
that the b-quark mass m (or the B-meson mass m (cid:39) 5.3GeV) is much larger than the
b B
QCD confinement scale Λ ∼ m − m ∼ 0.6GeV, and that all the decay products are
B b
much lighter than m . It is also important that the numerically dominant contribution
b
comes from processes where the b-quark decay and the photon emission occur practically
at the same point, i.e. at distances much smaller than 1/Λ.
Nonperturbativecorrectionsintheconsideredcasecanbestudiedusingtheframework
of HQET, as well as the Soft-Collinear Effective Theory (SCET). It is important to keep
in mind that some of the nonperturbative effects are not suppressed by Λ2/m2 but only
b,c
by Λ2/(m −2E )2 or Λ2/(m2−2m E ) (see, e.g., Ref. [62]). Such a behavior is precisely
b 0 b b 0
the reason of the above-mentioned growth of theoretical uncertainties when E tends to
0
4 known also as the spectator model rate
10